THE STRUCTURE OF SPIN SYSTEMS

William Arveson and Geoffrey Price

Department of Mathematics University of California Berkeley CA 94720, USA

Department of Mathematics U. S. Naval Academy Annapolis, MD 21402, USA

Abstract. A spin system is a sequence of self-adjoint unitary operators U_{1}, U_{2}, … acting on a Hilbert space H which either commute or anticommute, U_{i}U_{j} = ±U_{j}U_{i} for all i, j; it is is called irreducible when {U_{1}, U_{2}, … } is an irreducible set of operators. There is a unique infinite matrix (c_{ij}) with 0, 1 entries satisfying

U_{i}U_{j} = (−1)^{c_{ij}} U_{j}U_{i}, i, j = 1, 2, ….

Every matrix (c_{ij}) with 0, 1 entries satisfying c_{ij} = c_{ji} and c_{ii} = 0 arises from a nontrivial irreducible spin system, and there are uncountably many such matrices.

Infinite dimensional irreducible representations exist when the commutation matrix (c_{ij}) is of “infinite rank”. In such cases we show that the C^{∗}-algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor product of copies of M_{2}(ℂ), and we classify the irreducible spin systems associated with a given matrix (c_{ij}) up to approximate unitary equivalence. That follows from a structural result. The C^{∗}-algebra generated by the universal spin system u_{1}, u_{2}, … of (c_{ij}) decomposes into a tensor product C(X) ⊗ A, where X is a Cantor set (possibly finite) and A is either the CAR algebra or a finite tensor product of copies of M_{2}(ℂ).

The Bratteli diagram technology of AF algebras is not well suited to spin systems. Instead, we work out elementary properties of the ℤ_{2}-valued “symplectic” form

ω(x, y) = ∑_{p,q=1}^{∞} c_{pq}x_{q}y_{p},

x, y ranging over the free infninite dimensional vector space over the Galois field ℤ_{2}, and show that one can read off the structure of C(X) ⊗ A from properties of ω.

The authors acknowledge support from NSF grants DMS-9802474 and DMS-9706441.

1. Introduction.

A spin system is a sequence u_{1}, u_{2}, … of self-adjoint unitary elements of some unital C^{∗}-algebra which commute up to phase in the sense that

u_{i}u_{j} = λ_{ij}u_{k}u_{j}, i, j = 1, 2, …

where the λ_{ij} are complex numbers. Since u_{i}u_{j}u_{i}^{−1} = λ_{ij}u_{j} and u_{j}^{2} = 1, it follows that each λ_{ij} is −1 or +1. Thus there is a unique matrix of zeros and ones c_{ij} such that the commutation relations become

(1.1)  u_{i}u_{j} = (−1)^{c_{ij}}u_{j}u_{i},  i, j = 1, 2, ….

The matrix (c_{ij}) is symmetric and has zeros along the main diagonal. A concrete spin system U_{1}, U_{2}, ⋯ ⊆ B(H) is said to be irreducible when {U_{1}, U_{2}, … } is an irreducible set of operators. The purpose of this paper is to determine the structure of the C^{∗}-algebra generated by an irreducible spin system associated with a given 0-1 matrix (c_{ij}), and to classify such spin systems up to “approximate” unitary equivalence (Theorem C, section 4).

Quantum Spin Systems. Spin systems arise naturally in several contexts, including the theory of quantum spin systems ([BR], section 6.2), and in the theory of quantum computing (especially, systems involving a large or infinite number of qubits). For example, suppose we are given a mutually commuting sequence θ_{1}, θ_{2}, … of involutive ∗-automorphisms of B(H), i.e., θ_{j}^{2} = id, θ_{k}θ_{j} = θ_{j}θ_{k} for all j, k = 1, 2, … (one can imagine that θ_{k} represents reversing the state of a two-valued quantum observable located at the kth site). For each k one can find a unitary operator U_{k} such that θ_{k}(A) = U_{k}AU_{k}^{−1}, A ∈ B(H), and by replacing U_{k} with λU_{k} for an appropriate scalar λ if necessary, we can arrange that U_{k}^{2} = 1. Since θ_{i}θ_{j} = θ_{j}θ_{i} it follows that U_{i} and U_{j} must commute up to phase, hence there is a unique number c_{ij} ∈ {0, 1} such that (1.1) is satisfied. The matrix C = (c_{ij}) does not depend on the choices made and is therefore an invariant attached to the original sequence of automorphisms θ̄ = (θ_{1}, θ_{2}, …). The sequence θ̄ is ergodic in the sense that its fixed algebra is ℂ · 1 if, and only if, the spin systems Ū = (U_{1}, U_{2}, …) associated with it are irreducible.

Remarks on rank. Consider the commutation matrix (c_{ij}) associated with a spin system (1.1). If all coefficients c_{ij} vanish then C^{∗}(u_{1}, u_{2}, …) is commutative. More generally, C^{∗}(u_{1}, u_{2}, …) degenerates whenever (c_{ij}) is of finite rank, where the rank is defined as follows. Considering ℤ_{2} = {0, 1} as the two-element Galois field we may consider vector spaces over ℤ_{2}, and in particular we can form the free infinite dimensional vector space Γ = ℤ_{2} ⊕ ℤ_{2} ⊕ … over ℤ_{2}. Elements of Γ are sequences x = (x_{1}, x_{2}, …), x_{k} ∈ ℤ_{2}, which vanish eventually. The dual of Γ is identified with the vector space ℤ_{2}^{∞} of all sequences y = (y_{1}, y_{2}, …), y_{k} ∈ ℤ_{2}. The commutation matrix (c_{ij}) gives rise to a linear operator C : Γ → ℤ_{2}^{∞} by way of (Cx)_{k} = ∑_{j=1}^{∞} c_{kj}x_{j}, k = 1, 2, . . .. The rank of the matrix is defined by

rank (c_{ij}) = dim(CΓ).

Actually, what we have defined is the column rank of the matrix (c_{ij}), but because (c_{ij}) is a symmetric matrix its column and row ranks are the same. We will see below that the rank is finite iff the center of C^{∗}(u_{1}, u_{2}, …) is of finite codimension in C^{∗}(u_{1}, u_{2}, …) iff every irreducible spin system satisying (1.1) acts on a finite dimensional Hilbert space. Thus we are primarily concerned with the nondegenerate cases in which the commutation matrix (c_{ij}) is of infinite rank.

Remarks on Existence and Universality. A sequence u_{1}, u_{2}, … of unitary operators satisfying a given set of noncommutative equations f_{k}(u_{1}, u_{2}, …) = 0, k = 1, 2, … (we leave the precise nature of the noncommutative polynomials f_{k} unspecified) is said to be universal if every sequence U_{1}, U_{2}, ⋯ ∈ B(H) of concrete unitary operators that satisfies the equations can be obtained from it via a representation π : C^{∗}(u_{1}, u_{2}, …) → B(H) such that π(u_{k}) = U_{k}, k = 1, 2, . . ..

Of course, for bad choices of f_{k} (such as f_{k}(x_{1}, x_{2}, …) = x_{k}) there may be no unitary solutions to the set of equations except on the trivial Hilbert space H = {0}. But in all cases there is a universal solution...the direct sum of all concrete unitary solutions. Any two universal solutions (u_{1}, u_{2}, …) and (v_{1}, v_{2}, …) are equivalent in the sense that there is a unique ∗-isomorphism θ : C^{∗}(u_{1}, u_{2}, …) → C^{∗}(v_{1}, v_{2}, …) satisfying θ(u_{k}) = v_{k} for every k. Thus the C^{∗}-algebra generated by a universal sequence of solutions to the given set S of equations is uniquely determined by S.

Given an arbitrary matrix C = (c_{ij}) of zeros and ones satisfying the consistency requirements c_{ij} = c_{ji} and c_{jj} = 0 for all i, j = 1, 2, …, we consider the C^{∗}-algebra A_{C} = C^{∗}(u_{1}, u_{2}, …) generated by a universal spin system satisfying (1.1). The set of distinct matrices (c_{ij}) satisfying these conditions is of cardinality 2^{ℵ_{0}}, and each of them is associated with a nontrivial spin system (1.1) (see Proposition (1.1)). We determine the structure of these C^{∗}-algebras A_{C} in Theorem B, section 3.

Spin Systems in Characteristic p. We have found it helpful, even simplifying, to consider the natural generalization of spin systems to characteristic p where p is an arbitrary prime. By a spin system in characteristic p we mean a sequence of unitary operators u_{1}, u_{2}, … which are pth roots of unity in the sense that u_{j}^{p} = 1 for every j, and which satisfy commutation relations of the form

(1.2)  u_{i}u_{j} = ζ^{c_{ij}}u_{j}u_{i},  i, j = 1, 2, …,

where ζ = e^{2πi/p}, and where c_{ij} ∈ {0, 1, …, p − 1} = ℤ_{p}. The matrix (c_{ij}) is uniquely determined by (1.2). If we regard ℤ_{p} as a finite field in the usual way, then the matrix is skew-symmetric in that c_{ij} = −c_{ji} for every i, j = 1, 2, . . ..

The reason for considering the cases p > 2 can be clearly seen when one specializes the previous paragraph to p = 2. Indeed, a skew-symmetric matrix over the two-element field ℤ_{2} is the same as a symmetric matrix with zeros along the main diagonal. We found that viewing (c_{ij}) as a skew-symmetric matrix led in the right direction, whereas viewing it as a symmetric matrix with zeros along the diagonal led nowhere. Thus the case p = 2 can be misleading, and for that reason we consider the more general case of spin systems (1.2) in characteristic p.

Fixing a prime p, suppose we are given a skew-symmetric matrix (c_{ij}) of elements of the Galois field {0, 1, …, p − 1} = ℤ_{p}. Since ℤ_{p} is a field, we can form the free infinite dimensional vector space Γ over ℤ_{p}; elements of Γ are sequences x = (x_{1}, x_{2}, …) of elements of ℤ_{p} satisfying x_{k} = 0 for all but a finite number of k. The coefficients c_{ij} give rise to a bilinear form ω : Γ × Γ → ℤ_{p} by way of

(1.3)  ω(x, y) = ∑_{i,j=1}^{∞} c_{ij}x_{j}y_{i},  x, y ∈ Γ.

This bilinear form is skew-symmetric in that it satisfies ω(x, y) = −ω(y, x) for all x, y ∈ Γ, and it will occupy a central position throughout the sequel. The structure of such forms is described in Theorem A, section 2, and its Corollary.

Consider now the C^{∗}-algebra A generated by a sequence of unitary elements u_{1}, u_{2}, … satisfying u_{k}^{p} = 1 and the commutation relations (1.2). A word is a finite product of elements from {u_{1}, u_{2}, … }, and it is convenient to regard the identity 1 as the empty word. The set of linear combinations of words is a dense ∗-subalgebra of A which contains 1. Using the commutation relations (1.2), every word can be written in the form λu_{1}^{n_{1}}u_{2}^{n_{2}} … u_{r}^{n_{r}} where λ is a complex scalar. Thus we may use the elements of Γ to parameterize a spanning set of words as follows,

w_{x} = u_{1}^{x_{1}}u_{2}^{x_{2}} …, x = (x_{1}, x_{2}, …) ∈ Γ,

and one finds that

(1.4)  w_{x}w_{y} = ζ^{ω(x,y)}w_{y}w_{x},  x, y ∈ Γ,

where ζ = e^{2πi/p} and ω : Γ × Γ → ℤ_{p} is the bilinear form (1.3).

We will occasionally make use of a second bilinear form Q : Γ × Γ → ℤ_{p},

(1.5)  Q(x, y) = ∑_{1≤i<j} c_{ij}x_{j}y_{i},  x, y ∈ Γ.

Q is related to ω by ω(x, y) = Q(x, y) − Q(y, x), and it obeys the “Weyl” relations

(1.6)  w_{x}w_{y} = ζ^{Q(x,y)}w_{x+y},  x, y ∈ Γ.

We conclude the introduction with an observation about the existence of solutions of (1.2) for arbitrary coefficient matrices (c_{ij}). For p = 2, this generalizes the examples of finite dimensional spin systems described in [Bi].

Proposition 1.1. Let p = 2, 3, … be a prime and let (c_{ij}) be an arbitrary skewsymmetric matrix over the Galois field ℤ_{p} = {0, 1, …, p−1}. There a Hilbert space H ≠ {0} and a sequence of unitary operators U_{1}, U_{2}, ⋯ ∈ B(H) such that U_{k}^{p} = 1 and U_{j}U_{k} = ζ^{c_{jk}}U_{k}U_{j} for every j, k = 1, 2, …, where ζ = e^{2πi/p}.

proof. Regarding ℤ_{p} as an additive abelian group, consider the unitary operators S, V defined on the p-dimensional Hilbert space ℓ^{2}(ℤ_{p}) by

Sf(k) = f(k + 1), V f(k) = ζ^{k}f(k), f ∈ ℓ^{2}(ℤ_{p}), k ∈ ℤ_{p}.

We have S^{p} = V^{p} = 1, SV = ζV S, and in fact SV^{k} = ζ^{k}V^{k}S for all k ∈ ℤ.

Consider the L^{2}-space of the compact abelian group G = ℤ_{p} × ℤ_{p} × . . .. We can realize L^{2}(G) as the infinite tensor product of copies of ℓ^{2}(ℤ_{p}) along the stabilizing vector u ∈ ℓ^{2}(ℤ_{p}) where u is the constant function u(k) = 1, k ∈ ℤ_{p}. Thus for any finite sequence A_{1}, …, A_{r} of operators on ℓ^{2}(ℤ_{p}) we can form the operator

A_{1} ⊗ ⋯ ⊗ A_{r} ⊗ 1 ⊗ 1 ⊗ ⋯ ∈ B(L^{2}(G)).

The unitary operators U_{1}, U_{2}, … are defined on L^{2}(G) in terms of the given coefficients c_{ij} as follows; U_{1} = S ⊗ 1 ⊗ 1 ⊗ … and for k = 2, 3, …

U_{k} = V^{c_{1k}} ⊗ ⋯ ⊗ V^{c_{k−1k}} ⊗ S ⊗ 1 ⊗ 1 ⊗ ….

One can verify that U_{k}^{p} = 1, and U_{j}U_{k} = ζ^{c_{jk}}U_{k}U_{j} for 1 ≤ k < j.

As the preceding remarks on Quantum Spin Systems show, commutation relations of the form uv = λvu where λ ∈ 𝕋 arise whenever one considers commuting ∗-automorphisms of B(H), and in fact many natural contexts lead to projective representations of groups involving similar commutation relations. For example, they are associated with ergodic actions of compact groups on C^{∗}-algebras (see [BE], [J] and references therein). Since commutation relations of this type are so ubiquitous, we have made no effort to compile references to the related literature, even for the case of spin systems. Finally, we point out that the results of this paper generalize certain results in [PP], [P1], [P2] which concern spin systems for which the commutation matrix depends only on the separation c_{ij} = f(i − j).

2. Symplectic forms in characteristic p.

In this section we work out the symplectic linear algebra that underlies the results described above. While the results (and methods) are quite elementary, we have been unable to find what we require in the literature; indeed, fields of characteristic 2 are excluded from most treatments of linear algebra. Thus we provide a complete discussion. Throughout, F denotes a field, the primary cases being the Galois field F = ℤ_{p} of characteristic p where p is any prime including 2. Γ denotes the free infinite dimensional vector space over F, consisting of all sequences x = (x_{1}, x_{2}, …) of elements x_{k} ∈ F satisfying x_{k} = 0 for all but a finite number of k. The addition and scalar multiplication are defined pointwise,

x + y = (x_{1} + y_{1}, x_{2} + y_{2}, …), λ · x = (λx_{1}, λx_{2}, …)

λ being an element of F. A vector space V over F is said to be countably generated if it contains a sequence v_{1}, v_{2}, … such that every element of v is a finite linear combination of elements of {v_{1}, v_{2}, … }. For every countably generated vector space V over F there is a linear map L : Γ → V such that V = LΓ.

We are concerned with skew-symmetric bilinear forms B : Γ × Γ → F. The kernel of such a bilinear form is the subspace K = {x ∈ Γ : B(x, Γ) = {0}}. B is called a symplectic form when it is skew-symmetric and has kernel {0}, and a symplectic vector space is a pair (V, B) consisting of a countably generated vector space V over F and a symplectic bilinear form B : V × V → F. Two symplectic vector spaces (V, B) and (V′, B′) are congruent if there is a linear isomorphism L : V → V′ satisfying B′(Lx, Ly) = B(x, y) for all x, y ∈ V.

Any skew-symmeteric bilinear form defined on a vector space B : V × V → F gives rise to a symplectic vector space as follows. Letting K be the kernel of B, B promotes natrually to a bilinear form ω : V/K × V/K → F,

(2.1)  ω(x + K, y + K) = B(x, y),  x, y ∈ V.

(V/K, ω) is a symplectic vector space, and it is the trivial symplectic vector space only when B = 0.

Definition. The rank of B is the dimension of the vector space V/ ker B.

The rank of B is a nonnegative integer or ∞. We will see presently that when it is finite it must be an even integer n = 2r, r = 1, 2, . . ..

Further remarks on rank. Let C = (c_{ij}) be the commutation matrix associated with the relations (1.1). We have given a different definition of rank in the introduction, and we want to point out that the rank defined in the introduction is the same as the rank of the skew-symmetric form ω associated to it by (1.3). To see that consider the linear map L : Γ → ℤ_{2}^{∞} defined by

Lx = (ω(u_{1}, x), ω(u_{2}, x), …),

where u_{1}, u_{2}, … is the usual sequence of basis vectors in Γ, u_{k}(j) = δ_{kj}. Noting that the kth component of Lx is ω(u_{k}, x) = ∑_{j} c_{kj}x_{j}, one sees that the range of L is the linear span of the columns of (c_{ij}) and hence its dimension is the rank of the matrix (c_{ij}). On the other hand, the kernel of L is exactly ker ω, so that rank C = dim LΓ = dim(Γ/ ker ω), as asserted.

Let (V, ω) be a symplectic vector space. By a symplectic basis for V we mean a pair of sequences e_{1}, e_{2}, …, f_{1}, f_{2}, ⋯ ∈ V with the properties

(2.2)  ω(e_{i}, e_{j}) = 0, ω(f_{i}, f_{j}) = 0, ω(e_{i}, f_{j}) = δ_{ij},

for all i, j = 1, 2, … and which span V in the sense that every element of V is a finite linear combination of the elements {e_{i}, f_{j}}. The sequences are allowed to be either finite or infinite, but if one of them is finite then the other is also finite of the same length. A simple argument shows that any finite set of 2r vectors e_{1}, …, e_{r}, f_{1}, …, f_{r} which satisfy the relations (2.2) must be linearly independent. Thus a symplectic basis for V is a countable basis, and in particular V must be countably generated.

The Standard Examples. We describe the standard models of symplectic vector spaces of dimension n = 2, 4, 6, …, ∞ over an arbitrary field F. Consider first the case n = ∞. Let Γ = F ⊕ F ⊕ … be the vector space of all infinite sequences x = (x_{1}, x_{2}, …), where x_{k} ∈ F and x_{k} = 0 for all but a finite number of k. The symplectic space (V_{∞}, ω_{∞}) is defined by V_{∞} = Γ ⊕ Γ and

ω_{∞}((x, y), (x′, y′)) = ∑_{k=1}^{∞} y_{k}x′_{k} − x_{k}y′_{k}.

(V_{∞}, ω_{∞}) is a countably generated infinite dimensional symplectic vector space, and it has a natural symplectic basis {e_{j}, f_{k}}, defined by

e_{k} = (u_{k}, 0), f_{k} = (0, u_{k}), k = 1, 2, …

where u_{k} is the standard unit vector u_{k}(j) = δ_{kj}.

For n = 2r finite, we take V_{n} to be the 2r dimensional subspace F^{r} ⊕ F^{r} ⊆ V_{∞} and define ω_{n} by restricting ω_{∞} to V_{n}.

The following result implies that any two countably generated symplectic vector spaces of the same dimension are congruent.

Theorem A. Let F be a field of arbitrary characteristic.

A1 Every countably generated symplectic vector space (V, ω) over F has a symplectic basis. When the dimension of V is finite it must be even, dim V = 2r, r = 1, 2, . . ..

A2 Let ω be a skew-symmetric bilinear form on a countably generated vector space V, let K be the kernel of ω and let L be any vector space complement V = K ⊕ L. Then the restriction ω_{L} of ω to L is a symplectic form. If L′ is any other complement V = K ⊕ L′, then the symplectic spaces (L, ω_{L}) and (L′, ω_{L′}) are congruent.

proof of (A1). We first treat the simple case in which V is finite dimensional and nonzero. Choose any vector e_{1} ≠ 0 in V. By nondegeneracy, there is a vector f_{1} ∈ V with ω(e_{1}, f_{1}) = 1. In order to continue inductively, we require

Lemma 2.1. Let (V, ω) be a finite dimensional symplectic vector space, let S ⊆ V be a subspace such that the restriction of ω to S × S is nondegenerate, and let K be its symplectic complement

K = {x ∈ V : ω(x, S) = {0}}.

Then V = S ⊕ K.

proof. Obviously, S∩K = {0} because the restriction of ω to S×S is nondegenerate. We have to show that V = S + K, and since the intersection of these two spaces is trivial it suffices to show that dim S + dim K = dim V.

Assuming S ≠ {0}, let v_{1}, …, v_{r} be a basis for S, and consider the linear map L : V → F^{r} defined by

Lx = (ω(x, v_{1}), …, ω(x, v_{r})), x ∈ V.

The kernel of L is K, and we claim that LV = F^{r}. To prove that we show that the only linear functional f : F^{r} → F that vanishes on LV is f = 0. Indeed, writing

f(t_{1}, …, t_{r}) = ∑_{k=1}^{r} λ_{k}t_{k},

for certain λ_{j} ∈ F, the vector v = ∑_{k} λ_{k}v_{k} ∈ S satisfies ω(x, v) = f(Lx) = 0 for all x ∈ V. Since ω is nondegenerate we must have v = 0, hence λ_{1} = ⋯ = λ_{r} = 0, hence f = 0. We conclude that

dim V = dim ran L + dim ker L = dim F^{r} + dim K = dim S + dim K,

since dim S = r = dim F^{r}.

Inductively, suppose we have vectors e_{1}, …, e_{r}, f_{1}, …, f_{r} ∈ V which satisfy the symplectic requirements (2.2) insofar as they make sense, and let S be the subspace of V spanned by {e_{k}, f_{j} : 1 ≤ j, k ≤ r}. Since {e_{k}, f_{j}} is a symplectic basis for the restriction of ω to S ×S, the latter must be nondegenerate. By Lemma 2.1 we have V = S + K where K = {x ∈ V : ω(x, S) = {0}}. Thus we can choose a nonzero vector e_{r+1} in K. Since ω(e_{r+1}, S) = {0} and V = S + K, there must be a vector f_{r+1} ∈ K for which ω(e_{r+1}, f_{r+1}) = 1. An inductive argument completes the proof in the case where V is finite dimensional.

Remark. Notice that the preceding argument implies that in a finite dimensional symplectic vector space (V, ω), any set of vectors e_{1}, …, e_{r}, f_{1}, …, f_{r} ∈ V, which satisfy the relations (2.2), can be enlarged to a symplectic basis for V. It also shows that a finite dimensional symplectic vector space over an arbitrary field has even dimension 2 · r, r = 1, 2, . . ..

Turning now to the infinite dimensional case, we claim that there is an increasing sequence of finite dimensional subspaces E_{1} ⊆ E_{2} ⊆ ⋯ ⊆ V with ∪_{n}E_{n} = V, such that the restriction of ω to E_{n} × E_{n} is nondegenerate for every n. Suppose for the moment that this has been established. The preceding paragraphs show that we can find a symplectic basis for E_{1}. Since the restriction of ω to E_{2} × E_{2} is a symplectic form on E_{2}, the preceding remark implies that this symplectic set can be enlarged to a symplectic basis for E_{2}. Continuing inductively, we obtain an increasing sequence of symplectic sets, each one being a basis for its corresponding linear span E_{n}, n = 1, 2, …, and their union is a symplectic basis for ∪_{n}E_{n} = V.

Thus we have reduced the proof of (A1) to showing that there is such a sequence E_{1} ⊆ E_{2} ⊆ . . ..

Lemma 2.2. Let (V, ω) be a symplectic vector space and let E be a finite dimensional subspace of V. Then there is a subspace E′ ⊇ E of dimension at most 2 · dim E such that the restriction of ω to E′ × E′ is nondegenerate.

proof. Let K = {x ∈ E : ω(x, E) = 0} be the kernel of the restriction of ω to E ×E, and let k_{1}, …, k_{r} be a basis for K. We claim that there are vectors ℓ_{1}, …, ℓ_{r} ∈ V such that

(2.3)  ω(k_{i}, ℓ_{j}) = δ_{ij},  1 ≤ i, j ≤ r.

To see that, consider the r-dimensional vector space F^{r} = {(t_{1}, …, t_{r}) : t_{i} ∈ F }, and consider the linear map L : V → F^{r} defined by

L(x) = (ω(k_{1}, x), ω(k_{2}, x), …, ω(k_{r}, x)), x ∈ V.

We have to show that L is onto: L(V) = F^{r}. To prove that, we show that the only linear functional f : F^{r} → F which vanishes on the range of L is the zero functional. Choosing such an f, we can write

f(t_{1}, …, t_{r}) = λ_{1}t_{1} + ⋯ + λ_{r}t_{r}

for a unique r-tuple of scalars λ_{k} ∈ F. Since f(L(x)) = 0 for all x ∈ V we have

ω(∑_{j=1}^{r} λ_{j}k_{j}, x) = ∑_{j=1}^{r} λ_{j}ω(k_{j}, x) = f(L(x)) = 0.

By nondegeneracy, we must have ∑_{j} λ_{j}k_{j} = 0, hence λ_{1} = ⋯ = λ_{r} = 0 because k_{1}, …, k_{r} are linearly independent, thus (2.3) is proved.

Setting L = span{ℓ_{1}, …, ℓ_{r}}, notice that (2.3) implies that the restriction of ω to K × L is nondegenerate in the sense that for every k ∈ K,

(2.4)  ω(k, ℓ) = 0, for all ℓ ∈ L ⟹ k = 0,

while for every ℓ ∈ L,

(2.5)  ω(k, ℓ) = 0, for all k ∈ K ⟹ ℓ = 0.

Choose such a set of vectors ℓ_{1}, …, ℓ_{r} ∈ V, let L = span{ℓ_{1}, …, ℓ_{r}}, and define E′ = E + L. We show that the restriction of ω to E′ × E′ is nondegenerate. For that, suppose that z ∈ E′ has the property that ω(z, z′) = 0 for every z′ ∈ E′. We can write z = x + ℓ where x ∈ E and ℓ ∈ L . Then

ω(z, z′) = ω(x, z′) + ω(ℓ, z′) = 0

for all z′ ∈ E′. Picking z′ ∈ K and noting that ω(x, K) = {0} (by definition of K), we conclude that ω(ℓ, z′) = 0 for all z′ ∈ K. Because of (2.5), we conclude that ℓ = 0. Hence ω(x, E′) = {0}. Since x ∈ E ⊆ E′ this implies that x is an element of K for which ω(x, E′) = 0. By (2.4) this implies x = 0.

The proof of (A1) is completed as follows. Since V is countably generated there is a spanning sequence of nonzero vectors v_{1}, v_{2}, ⋯ ∈ V; we will construct an increasing sequence E_{n} of finite dimensional subspaces such that E_{n} contains v_{1}, …, v_{n} and the restriction of ω to E_{n} is nondegenerate. Since v_{1} ≠ 0 and ω is nondegenerate, choose any w ∈ v such that ω(v_{1}, w) = 1, and set E_{1} = span{v_{1}, w}. The restriction of ω to E_{1} is nondegenerate because {v_{1}, w} is a symplectic basis.

Suppose now that we have finite dimensional subspaces E_{1} ⊆ ⋯ ⊆ E_{n} such that E_{k} contains v_{1}, …, v_{k} and the restriction of ω to each E_{k} is nondegenerate. Applying Lemma 2.2 to the space spanned by E_{n} and v_{n+1}, we find a finite dimensional space E_{n+1} containing v_{n+1} and E_{n} such that the restriction of ω to E_{n+1} × E_{n+1} is nondegenerate. An induction completes the proof of (1).

In order to prove (A2), consider the natural symplectic space (V/K, ω) described above. We claim that for every subspace L of V satisfying L ∩ K = {0} and L + K = V, the symplectic spaces (L, ω_{L}) and (V/K, ω) are congruent; i.e., there is a linear isomorphism T : L → V/K such that

(2.6)  ω(T x, T y) = B(x, y) = ω_{L}(x, y),  x, y ∈ L,

where ω_{L} is the restriction of B to L×L. To see that, define T x = x+K, x ∈ L. T is a linear isomorphism because L is a complement of K, and (2.6) follows because for any x, y ∈ V we have ω(x + K, y + K) = B(x, y) by definition of ω, so when x, y ∈ L we have (2.6).

For any other subspace L′ with V = K ⊕ L′, (L′, ω_{L′}) is also congruent to (V/K, ω), hence it is congruent to (L, ω_{L}).

Corollary. Any two countably generated symplectic vector spaces of the same dimension n = 2r, r = 1, 2, …, ∞ are congruent.

proof. Let (V, ω) be a symplectic vector space of dimension n = 2r, r = 1, 2, …, ∞. By Theorem A, we can find a (finite or infinite) symplectic basis {e_{k}, f_{j}} for V, and once we have that there is an obvious way to transform (V, ω) congruently to the standard example (V_{n}, ω_{n}).

Examples of commutation matrices. The above results have concrete implications about how to exhibit sequences of unitary operators that generate the infinite dimensional CAR algebra; they also provide a systematic method for generating all possible skew-symmetric matrices C = (c_{ij}) with entries in ℤ_{2} which are nondegenerate in the sense that their associated bilinear forms

(2.7)  ω_{C}(x, y) = ∑_{i,j=1}^{∞} c_{ij}x_{j}y_{i},  x, y ∈ Γ

have trivial kernel. We abuse terminology slightly by calling such a matrix C symplectic. Starting with any countably infinite symplectic vector space (V, ω) over ℤ_{2}, such as the standard example (V_{∞}, ω_{∞}) described above, let v_{1}, v_{2}, … be any basis for V and define

c_{ij} = ω(v_{i}, v_{j}), i, j = 1, 2, ….

One verifies directly that C = (c_{ij}) is a symplectic matrix. Moreover, the Corollary of Theorem A implies that every symplectic matrix arises in this way from some basis v_{1}, v_{2}, … for V.

One can view this construction in more concrete operator-theoretic terms by making use of the standard self-adjoint generators of the CAR algebra as follows. Consider the Clifford algebra C generated by an infinite sequence W_{1}, W_{2}, … of unitary operators satisfying

W_{i}W_{j} + W_{j}W_{i} = 2δ_{ij}1, i, j = 1, 2, ….

Since W_{i} and W_{j} anticommute when i ≠ j, the commutation matrix A = (a_{ij}) associated with a Clifford sequence is

a_{ij} = \{ \begin{matrix}1, & i ≠ j, \\ 0, & i = j, \end{matrix}

and its associated form is

ω_{A}(x, y) = ∑_{p≠q} x_{q}y_{p} = (∑_{k} x_{k})(∑_{k} y_{k}) − ∑_{k} x_{k}y_{k}.

One verifies easily that ω_{A} is nondegenerate. Choosing an arbitrary basis v_{1}, v_{2}, … for Γ, we obtain the most general symplectic matrix C = (c_{ij}) as follows

(2.8)  c_{ij} = ω_{A}(v_{i}, v_{j}) = ∑_{p≠q} v_{i}(q)v_{j}(p).

Each element v_{k} in this basis is associated with a word in the original sequence (W_{n}), namely U_{k} = W_{1}^{v_{k}(1)}W_{2}^{v_{k}(2)} . . .. The unitary operators U_{1}, U_{2}, … satisfy

(2.9)  U_{i}U_{j} = (−1)^{c_{ij}}U_{j}U_{i}  i, j = 1, 2, …

and, after multiplication by suitable phase factors, U_{1}, U_{2}, … becomes a spin system which generates the Clifford algebra C.

3. The Universal C^{∗}-algebra.

The purpose of this section is to prove Theorem B. Let p = 2, 3, … be a prime and let u_{1}, u_{2}, … be a universal sequence of unitary operators satisfying u_{k}^{p} = 1 for all k and the commutation relations (1.2). Let ω : Γ × Γ → ℤ_{p} be the skew-symmetric form (1.3) and let n = 2r be its rank, r = 1, 2, …, ∞.

Then C^{∗}(u_{1}, u_{2}, …) is isomorphic to C(X)⊗B, where X is a totally disconnected compact metrizable space, and where B = M_{p^{r}}(ℂ) if r is finite and is a UHF algebra of type p^{∞} if r = ∞.

The center C(X)⊗1 is the closed linear span of the set of words {w_{x} : x ∈ ker ω}. C^{∗}(u_{1}, u_{2}, …) is simple iff its center is trivial iff ω is a symplectic form.

Remarks. Since every quotient of C(X) for X a compact totally disconnected metrizable space is of the form C(Y) for Y of the same type, it follows that any sequence of unitary operators U_{1}, U_{2}, … that satisfies U_{k}^{p} = 1 and the relations (1.2), whether it is universal or not, must generate a C^{∗}-algebra of the same general type C(Y) ⊗ B as the universal one C(X) ⊗ B. If {U_{1}, U_{2}, … } is irreducible and ω is of infinite rank, then X reduces to a point and C^{∗}(U_{1}, U_{2}, …) is a UHF algebra of type p^{∞}.

Before giving the proof of Theorem B, we require two elementary results.

Lemma 3.1. Let A be a unital C^{∗}-algebra which is generated by two mutually commuting unital C^{∗}-subalgebras Z, B with the properties

(i) Z ≅ C(X) is commutative, and (ii) B is a UHF algebra.

Then Z is the center of A and A ≅ C(X) ⊗ B.

proof of Lemma 3.1. The proof is straightforward and we merely sketch the argument. Suppose first that the subalgebra B is finite dimensional, hence isomorphic to the matrix algebra M_{n}(ℂ) for some n = 1, 2, . . .. Pick a set of matrix units e_{ij}, 1 ≤ i, j ≤ n for B. Thus e_{ij}e_{kl} = δ_{jk}e_{il}, e_{ij}^{∗} = e_{ji}, and e_{11} + ⋯ + e_{nn} = 1. Using these relations and the fact that the elements of Z commute with the e_{ij} one finds that for arbitrary z_{ij} ∈ Z, 1 ≤ i, j ≤ n,

∑_{i,j=1}^{n} z_{ij}e_{ij} = 0 ⟹ z_{ij} = 0, for all 1 ≤ i, j ≤ n.

Thus if we consider Z ⊗ B to be the C^{∗}-algebra M_{n}(Z) then the preceding observation shows that the natural ∗-homomorphism π : M_{n}(Z) → A defined by

π((z_{ij})) = ∑_{i.j=1}^{n} z_{ij}e_{ij}

is injective; it also has dense range, hence it is a ∗-isomorphism which carries the center of M_{n}(Z) onto Z.

In the general case, B is the norm closure of an increasing sequence of algebras B_{n} of the above type. The preceding argument shows that the natural surjective ∗-homomorphism π : Z ⊗B → A restricts to an isometric ∗-homomorphism on each Z ⊗ B_{n}, hence it is an isometric ∗-isomorphism.

Lemma 3.2. Let p be a positive integer and let V and W be unitary operators in some C^{∗}-algebra satisfying V^{p} = W^{p} = 1 and V W = ζW V where ζ = e^{2πi/p}. Then C^{∗}(V, W) ≅ M_{p}(ℂ).

proof of Lemma 3.2. Since W^{p} = 1, the spectrum σ(W) of W is contained in the set of pth roots of unity, and because V W V^{−1} = ζW, σ(W) is invariant under multiplication by ζ. Hence σ(W) = {1, ζ, ζ^{2}, …, ζ^{p−1}}. Letting P_{k} be the spectral projection of W corresponding to the eigenvalue ζ^{k}, k = 0, 1, …, p−1, the commutation relation V^{i}W V^{−i} = ζ^{i}W^{j} implies that V^{i}P_{j} = P_{i+j}V^{i}, where the sum i + j is interpreted modulo p. Together with V^{p} = 1, this implies that the operators e_{ij} = V^{i−j}P_{j}, 0 ≤ i, j ≤ p − 1, are a set of p × p matrix units which have C^{∗}(V, W) as their linear span.

proof of Theorem B. Fix a universal sequence u_{1}, u_{2}, … as above and let Z be the closed linear span of the words of the form w_{x} = u_{1}^{x_{1}}u_{2}^{x_{2}} … where x ∈ ker ω. Notice that because w_{x}w_{y} = (−1)^{ω(x,y)}w_{y}w_{x}, it follows that every word w_{x} with x ∈ ker ω belongs to the center of C^{∗}(u_{1}, u_{2}, …). Note that for each x ∈ Γ we can choose a scalar λ_{x} ∈ 𝕋 with the property that (λ_{x}w_{x})^{p} = 1. It is possible to do this because the relation w_{s}w_{t} = ζ^{Q(s,t)}w_{s+t} implies that w_{x}^{p} is a scalar multiple of w_{px} = w_{0} = 1. One can specify λ_{x} explicitly, but it is not necessary to do so. Thus Z is a commutative AF algebra isomorphic to C(X) for X a compact metrizable totally disconnected space. Because of Lemma 3.1, it is enough to show that there is a UHF algebra B ⊆ C^{∗}(u_{1}, u_{2}, …) of the asserted type such that C^{∗}(u_{1}, u_{2}, …) is generated by Z ∪ B.

By Theorem A, Γ decomposes into a direct sum of vector spaces Γ = ker ω ⊕ L, where the restriction of ω to L × L is a symplectic form, and where dim L is the rank of ω. Since L is a vector space, the relation w_{x}w_{y} = ζ^{Q(x,y)}w_{x+y} implies that B = \overline{span}\{w_{x} : x ∈ L\} is a C^{∗}-subalgebra of A. Moreover, since Γ = ker ω + L, the set of products of words of the form w_{x}w_{y} = ζ^{Q(x,y)}w_{x+y}, x ∈ ker ω, y ∈ L have A as their closed linear span. Thus Z ∪ B generates A.

It remains to show that B is a UHF algebra of the asserted type. Suppose first that dim L = 2r is finite. By Theorem A we can find a symplectic basis e_{1}, …, e_{r}, f_{1}, …, f_{r} for the symplectic vector space (L, ω_{L}) obtained by restricting ω to L. Consider the operators V_{1}, …, V_{r}, W_{1}, …, W_{r} defined by

(3.1)  V_{k} = λ_{e_{k}}w_{e_{k}}, W_{k} = λ_{f_{k}}w_{f_{k}},

where the scalars λ_{x} are as above. Every x ∈ L is a linear combination of elements of e_{1}, …, e_{r}, f_{1}, …, f_{r}, hence the set of all products V_{1}^{m_{1}} … V_{r}^{m_{r}}W_{1}^{n_{1}} …, W_{r}^{n_{r}}, m_{1}, …, m_{r}, n_{1}, …, n_{r} = 0, 1, …, p − 1, spans B. We have already arranged that V_{k}^{p} = W_{k}^{p} = 1 for every k. Note that for all i, j = 1, …, r

(3.2)  V_{i}V_{j} = V_{j}V_{i}, W_{i}W_{j} = W_{j}W_{i}, V_{i}W_{j} = ζ^{δ_{ij}}W_{j}V_{i},

δ_{ij} denoting the Kronecker delta. Indeed, these relations are immediate consequences of the basic formula w_{x}w_{y} = ζ^{ω(x,y)}w_{y}w_{x} and the fact that {e_{i}, f_{j}} is a symplectic set for ω. It follows from (3.2) that the C^{∗}-algebras C^{∗}(V_{i}, W_{i}) and C^{∗}(V_{j}, W_{j}) commute for i ≠ j; and by Lemma 3.2 each C^{∗}(V_{k}, W_{k}) is isomorphic to M_{p}(ℂ). Thus B is isomorphic to a tensor product of r compies of M_{p}(ℂ).

If dim L is infinite, then another application of Theorem A provides an infinite symplectic basis e_{1}, e_{2}, …, f_{1}, f_{2}, … for L. We define V_{1}, V_{2}, …, W_{1}, W_{2}, … by (3.1) as before, and these operators satisfy (3.2). In this case, the C^{∗}-algebra B generated by V_{i}, W_{j} commutes with Z, and is generated by an increasing sequence of subalgebras B_{1} ⊆ B_{2} ⊆ …

B_{n} = C^{∗}(V_{1}, …, V_{n}, W_{1}, …, W_{n}), n = 1, 2, ….

The preceding paragraph shows that B_{n} is isomorphic M_{p^{n}}(ℂ). Hence B is a UHF algebra of type p^{∞}.

The assertions of the third paragraph of Theorem B are obvious consequences of what has already been proved.

4. Irreducible Spin Systems.

Let C = (c_{ij}) be a matrix of zeros and ones, fixed throughout the remainder of this section; in order to rule out the degeneracies described in the introduction, we also assume that (c_{ij}) is of infinite rank. Thus, the ℤ_{2}-valued bilinear form

(4.1)  ω(x, y) = ∑_{p,q=1}^{∞} c_{pq}x_{q}y_{p},  x, y ∈ Γ

associated with C = (c_{ij}) has the property that Γ/ ker ω is infinite dimensional, ker ω being the linear subspace {x ∈ Γ : ω(x, Γ) = {0}} ⊆ Γ.

The purpose of this section is to classify the irreducible spin systems associated with C. Thus we consider irreducible spin systems Ū = (U_{1}, U_{2}, …) acting on an infinite dimensional Hilbert space H, satisfying

(4.2)  U_{i}U_{j} = (−1)^{c_{ij}}U_{j}U_{i},  i, j = 1, 2 ….

Theorem B implies that C^{∗}(U_{1}, U_{2}, …) is the CAR algebra, and since the CAR algebra is a simple C^{∗}-algebra not of type I, there can be no meaningful classification of such sequences Ū up to unitary equivalence. The equivalence relation that is appropriate for irreducible spin systems is Voiculescu’s notion of approximate unitary equivalence [A2]. Two spin systems Ū and V̄, acting on infinite dimensional Hilbert spaces H and K, respectively, are said to be equivalent (written Ū ∼ V̄) if there is a sequence of unitary operators W_{1}, W_{2}, ⋯: H → K such that

lim_{n→∞} ∥W_{n}U_{k}W_{n}^{−1} − V_{k}∥ = 0 k = 1, 2, ….

We first introduce an invariant for irreducible spin systems Ū. For every x ∈ Γ there is a word

W_{x} = U_{1}^{x_{1}}U_{2}^{x_{2}} …, x ∈ Γ,

and we have W_{x}W_{y} = (−1)^{Q(x,y)}W_{x+y} for all x, y ∈ Γ, where Q : Γ × Γ → ℤ_{2} is the bilinear form (1.5). If x ∈ ker ω then by (1.4) W_{x} commutes with all words, and by irreducibility it must be a scalar multiple of the identity

W_{x} = f(x)1, x ∈ ker ω.

This defines a function f : ker ω → 𝕋 satisfying the functional equation

(4.3)  f(x)f(y) = (−1)^{Q(x,y)}f(x + y),  x, y ∈ ker ω.

f is called the standard invariant associated with the irreducible spin system Ū. Notice that (4.3) implies that f(0) = 1. Since f(x)^{2} = (−1)^{Q(x,x)}f(2x) = (−1)^{Q(x,x)}f(0) = ±1, it follows that f must take values in the multiplicative group of fourth roots of unity,

f(x)^{4} = 1, x ∈ ker ω.

Proposition 4.1. Let Ū = (U_{1}, U_{2}, …) and Ū′ = (U′_{1}, U′_{2}, …) be two irreducible spin systems on Hilbert spaces H, H′ which satisfy the relations (4.2), and let π, π′ be the representations of the universal C^{∗}-algebra A_{C} = C^{∗}(u_{1}, u_{2}, …) defined by π(u_{k}) = U_{k}, π′(u_{k}) = U′_{k}, k = 1, 2, . . .. The following are equivalent.

(i) Ū ∼ Ū′.

(ii) ker π = ker π′.

(iii) Ū and Ū′ have the same standard invariant.

(iv) For every n = 1, 2, …, there is a unitary operator W_{n} : H → H′ such that

W_{n}U_{k}W_{n}^{−1} = U′_{k}, k = 1, 2, …, n.

proof. The implications (i) ⟹ (ii) ⟹ (iii) and (iv) ⟹ (i) are straightforward. We prove (iii) ⟹ (iv). For that we will make use of the following elementary result. The proof, a straightforward exercise in elementary multiplicity theory (see [A1]), is omitted.

Lemma 4.1. Let B be a finite dimensional C^{∗}-algebra and let π_{1}, π_{2} be two faithful nondegenerate representations of B on Hilbert spaces H_{1}, H_{2} such that π_{j}(B)∩K_{j} = {0} for j = 1, 2, K_{j} denoting the compact operators on H_{j}. Then π_{1} and π_{2} are unitarily equivalent.

Let f and f′ be the respective standard invariants of Ū and Ū′. Assuming that f = f′ as in (iii), we have to verify (iv), and by replacing the spin system Ū′ with a unitarily equivalent one, we may assume that both Ū and Ū′ act on the same Hilbert space. Consider the words w_{x} = u_{1}^{x_{1}}u_{2}^{x_{2}} … in A_{C} corresponding to elements x ∈ ker ω. Since f = f′ we have

π(w_{x}) = U_{1}^{x_{1}}U_{2}^{x_{2}} ⋯ = f(x)1 = f′(x)1 = π′(w_{x}), x ∈ ker ω.

Since by Theorem B the central words of this type have the center Z of A_{C} as their closed linear span, it follows that π ↾_{Z}= π′ ↾_{Z}. Theorem B also implies that A_{C} is isomorphic to C(X) ⊗ C where C is the CAR algebra, hence any two irreducible representations that agree on the center must have the same kernel (corresponding to some point p ∈ X). Hence ker π = ker π′.

It follows that for every operator A ∈ A_{C} we have ∥π(A)∥ = ∥π′(A)∥, hence there is a unique ∗-isomorphism α : C^{∗}(U_{1}, U_{2}, …) → C^{∗}(U′_{1}, U′_{2}, …) such that α ∘ π = π′. Both of these are simple unital C^{∗}-algebras (they are isomorphic to the CAR algebra), and hence contain no nonzero compact operators. Noting that the restriction of α to C^{∗}(U_{1}, …, U_{n}) is a ∗-isomorphism onto C^{∗}(U′_{1}, …, U′_{n}) which carries the n-tuple of operators (U_{1}, …, U_{n}) to (U′_{1}, …, U′_{n}), an application of Lemma 4.1 implies that each of these restrictions is implemented by a unitary operator W_{n} ∈ B(H), and (iv) follows.

In conclusion, we describe how the irreducible spin systems associated with a commutation matrix C can be described and classified in terms of any one of them. For any irreducible spin system Ū = (U_{1}, U_{2}, …), we consider the spin systems that can be obtained from it by changing phases as follows. For every sequence of numbers γ = (γ_{1}, γ_{2}, …) in {0, 1} = ℤ_{2} consider the sequence of unitary operators

Ū^{γ} = ((−1)^{γ_{1}}U_{1}, (−1)^{γ_{2}}U_{2}, …).

It is clear that Ū^{γ} is an irreducible spin system satisfying the same commutation relations as Ū. We now show that these “phase shifted” versions of Ū provide all possible standard invariants.

Lemma 4.2. Let Ū be an irreducible spin system, let f : ker ω → 𝕋 be its standard invariant, and let g : ker ω → 𝕋 be any function satifying the same functional equation (4.3)

g(x)g(y) = (−1)^{Q(x,y)}g(x + y), x, y ∈ ker ω.

Then there is a γ = (γ_{1}, γ_{2}, …) ∈ ℤ_{2}^{∞} such that g is the standard invariant of Ū^{γ}.

proof. For γ ∈ ℤ_{2}^{∞}, we can express the standard invariant f^{γ} of Ū^{γ} in terms of the standard invariant f of Ū as follows. For every x ∈ Γ the word for Ū^{γ} is

(−1)^{∑_{k} γ_{k}x_{k}}U_{1}^{x_{1}}U_{2}^{x_{2}} …,

hence for x ∈ ker ω we have

(4.4)  f^{γ}(x) = (−1)^{∑_{k} γ_{k}x_{k}}f(x).

Now both g and f satisfy (4.3), hence the function h : ker ω → 𝕋 defined by h(x) = g(x)/f(x) satisfies

h(x + y) = h(x)h(y), x, y ∈ ker ω.

Notice too that since x + x = 0 for all x ∈ ker ω we have h(x)^{2} = h(x)h(x) = h(x + x) = h(0) = 1. It follows that h(x) = ±1 for all x ∈ ker ω. Thus there is a unique function θ : ker ω → {0, 1} = ℤ_{2} satisfying

(4.5)  g(x)/f(x) = h(x) = (−1)^{θ(x)},  x ∈ ker ω,

and we have θ(x + y) = θ(x) + θ(y) relative to the addition in the field ℤ_{2} because h(x + y) = h(x)h(y) for x, y ∈ ker ω.

We may consider θ : ker ω → ℤ_{2} as a linear functional defined on the vector space ker ω ⊆ Γ. A familiar argument implies that a linear functional defined on a subspace of a vector space can be extended to a linear functional defined on the entire space. Thus we may find a function θ̃: Γ → ℤ_{2} such that θ̃(x+y) = θ̃(x)+θ̃(y) for all x, y ∈ Γ and which restricts to θ on ker ω. Letting u_{1}, u_{2}, … be the usual basis of unit vectors for Γ, u_{k}(j) = δ_{kj}, we define γ = (γ_{1}, γ_{2}, …) ∈ ℤ_{2}^{∞} by γ_{k} = θ̃(u_{k}), k = 1, 2, . . .. For every x = (x_{1}, x_{2}, …) ∈ Γ we have θ̃(x) = ∑_{k=1}^{∞} θ̃(u_{k})x_{k} = ∑_{k=1}^{∞} γ_{k}x_{k}. Substituting the latter into (4.5) we find that

g(x) = (−1)^{θ(x)}f(x) = (−1)^{∑_{k=1}^{∞} γ_{k}x_{k}}f(x), x ∈ ker ω.

By (4.4), this is the standard invariant f^{γ} of Ū^{γ}.

Theorem C. Let Ū be any irreducible spin system satisfying the commutation relations (1.1) and let ω be the skew-symmetric form (1.3). Every irreducible spin sytem satisying the same commutation relations is equivalent to Ū^{γ} for some γ ∈ ℤ_{2}^{∞}. Given two sequences γ, γ′ in ℤ_{2}^{∞}, the spin systems Ū^{γ} and Ū^{γ′} are equivalent if and only if γ and γ′ define the same linear functional on ker ω,

∑_{k=1}^{∞} γ_{k}x_{k} = ∑_{k=1}^{∞} γ′_{k}x_{k}, x ∈ ker ω.

In particular, if ker ω is of finite dimension d as a vector space over ℤ_{2}, then there are exactly 2^{d} equivalence classes of irreducible spin systems associated with C. If ker ω is infinite dimensional then the set of distinct equivalence classes of irreducible spin systems has the cardinality of the continuum 2^{ℵ_{0}}.

proof. Fix an irreducible spin system Ū as above, and let γ and γ′ be two sequences in ℤ^{∞}. We show first that Ū^{γ} ∼ Ū^{γ′} ⟺

(4.6)  ∑_{k=1}^{∞}(γ_{k})x_{k} = ∑_{k=1}^{∞} γ′_{k}x_{k},  x ∈ ker ω.

Indeed, lettting f^{γ} and f^{γ′} be the standard invariants for Ū^{γ} and Ū^{γ′}, we see from (4.4) that

f^{γ}(x) = (−1)^{∑_{k} γ_{k}x_{k}}f(x), f^{γ′}(x) = (−1)^{∑_{k} γ′_{k}x_{k}}f(x), x ∈ ker ω

and hence f^{γ} = f^{γ′} iff (4.6) holds. By the characterization (iii) of Proposition (4.1), that is equivalent to Ū^{γ} ∼ Ū^{γ′}.

Now let V = (V_{1}, V_{2}, …) be an arbitrary irreducible spin system associated with C, and let g : ker ω → 𝕋 be its standard invariant. Lemma 4.2 implies that there is a γ ∈ ℤ_{2}^{∞} such that g = f^{γ}, and by part (iii) of Proposition 4.1 we conclude that V̄ ∼ Ū^{γ}.

It remains only to establish the results on cardinality, and in view of what has been proved, we simply have to count the distinct functions g : ker ω → 𝕋 that satisfy the functional equation (4.3). Letting f be the standard invariant of Ū, the proof of Lemma 2 shows that every such g is obtained from it by way of

g(x) = (−1)^{θ(x)}f(x), x ∈ ker ω,

where θ : ker ω → ℤ_{2} is a (necessarily unique) linear functional. Thus the set of all such g is in bijective correspondence with the set of linear functionals on ker ω. If ker ω is of finite dimension d then by choosing a basis e_{1}, …, e_{d} for ker ω we find that the set of all such θ is in bijective correspondence with the set of all functions from {e_{1}, …, e_{d}} to ℤ_{2}, and the cardinality of that set is 2^{d}.

If ker ω is infinite dimensional, then since it is a countably generated vector space it has a basis {e_{1}, e_{2}, … }. As in the preceding paragraph, the set of all standard invariants is in bijective correspondence with the set of all linear functionals on ker ω, which in turn corresponds bijectively with the set of all functions from {e_{1}, e_{2}, … } to ℤ_{2}, a set of cardinality 2^{ℵ_{0}}.

When the commutation matrix is symplectic one has the following uniqueness:

Corollary. Let C = (c_{ij}) be an infinite matrix of zeros and ones which is skewsymmetric and nondegenerate. Then any two irreducible spin systems satisfying the commutation relations U_{i}U_{j} = (−1)^{c_{ij}}U_{j}U_{i} are approximately unitarily equivalent.

Remark. In such cases the C^{∗}-algebra A_{C} associated with C is the CAR algebra, and is therefore simple not of type I. In view of Proposition 4.1, the Corollary remains valid verbatim if one deletes the irreducibility hypothesis.

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[P2] G. L. Price, Shifts on the hyperfinite II_{1} factor, J. Funct. Anal. 156 (1998), 121–169.